# Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories

## TL;DR

This paper investigates the Gribov ambiguity in topological Yang-Mills theories, showing that, unlike in non-topological cases, the Gribov problem does not affect the quantum theory due to the trivial gap equation and absence of radiative corrections.

## Contribution

It demonstrates that the Gribov copies are inoffensive in topological Yang-Mills theories and that the associated gap equation has only a trivial solution, unlike in standard Yang-Mills theories.

## Key findings

- Gribov copies are present but do not affect the quantum theory
- The gap equation admits only a trivial solution in this topological context
- Radiative corrections are absent, preventing the introduction of a Gribov mass parameter

## Abstract

We study the Gribov problem in four-dimensional topological Yang-Mills theories following the Baulieu-Singer approach in the (anti-)self-dual Landau gauges. This is a gauge-fixed approach that allows to recover the topological spectrum, as first constructed by Witten, by means of an equivariant (or constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this has repercussions for this analysis. The resolution of the small (infinitesimal) gauge copies, in general, affects the dynamics of the underlying theory. In particular, treating the Gribov problem for the standard Landau gauge condition in non-topological Yang-Mills theories strongly affects the dynamics of the theory in the infrared. In the current paper, although the theory is investigated with the same gauge condition, the effects of the copies turn out to be completely different. In other words: in both cases, the copies are there, but the effects are very different. As suggested by the tree-level exactness of the topological model in this gauge choice, the Gribov copies are shown to be inoffensive at the quantum level. To be more precise, following Gribov, we discuss the path integral restriction to the Gribov horizon. The associated gap equation, which fixes the so-called Gribov parameter, is however shown to only possess a trivial solution, making the restriction obsolete. We relate this to the absence of radiative corrections in both gauge and ghost sectors. We give further evidence by employing the renormalization group which shows that, for this kind of topological model, the gap equation indeed forbids the introduction of a massive Gribov parameter.